1.
The use of “quantifier” (German “Quantor”,
French “quantificateur”, etc.) to denote ∀ and
∃ became established in logic toward the end of the 1920s.

2.
As a rule of thumb we use a smaller sans-serif font for quantifier
expressions and italics for the signified quantifiers. In logical
languages, on the other hand, it is convenient to abuse notation
somewhat by using the same symbol for both the expression and the
quantifier, when no confusion results. We sometimes do the same for
predicate symbols, so that the letters A,
B,…, R,… stand for both the symbol and
the set or relation it denotes.

3.
See Parsons (2004) for an illuminating account of the Aristotelian
square of opposition and some modern misunderstandings about it.
Peters and Westerståhl (2006), ch. 1.1.1, has a quick comparison with the ‘modern square’, which differs from the Aristotelian one in that all doesn't have existential import but not
all does, and, more importantly, that the relations along this sides of the square are different. For a detailed account of these differences, and of squares of opposition generated by various generalized quantifiers, see Westerståhl (2011).

4.
See Peters and Westerståhl (2006), ch. 1.4, for a proof that
there does not exist a semantics assigning individuals or sets of
individuals even to phrases of the two forms all A and some B in a systematic (compositional) way.

5.
Some such properties were considered even in the early days of
predicate logic, for example, the quantifier ∃! meaning
‘there exists exactly one.’

6.
Rather than seeing generalized quantifiers as mappings from universes
to second-order relations, Lindström took them to be classes of
models of the corresponding type. This is a negligible difference,
since we have

QM(R1,…,
Rk) ⇔ (M,
R1,…, Rk)
∈ Q

(see section 6 for the notation on the right-hand side).

7.
It is often held that the idea of a totality of everything has been
shown to be incoherent by Russell's paradox. Indeed, the paradox
proved Frege's original system to be inconsistent, and shows that
there cannot be a set containing all sets. For a defense of the
notion of ‘everything’, and an argument that this notion
is not only coherent but in fact indispensable, see Williamson (2003).

8.
These are the modern variants. Aristotle considered all with existential import,
i.e.,

(allei)M(A,
B) ⇔ ∅ ≠ A ⊆
B

and similarly for not
all, which he took to be the negation of
allei.

9.
We are here interpreting most as ‘more than half of the’ (on
finite universes). Often, most rather seems to mean something like ‘a
large majority of the’. There is probably some vagueness
involved, as well as an ambiguity: the threshold may be
set at different values in different contexts. We are not dealing with
vagueness here, and we assume that a fixed context assumption
chooses a suitable meaning. By default, we thus set the
threshold to 1/2.

10.
A main source for the mathematics of model-theoretic logics is
Barwise and Feferman (1985).

11.
Here is a fact which is non-trivial but still relatively easy to
prove:

FO(MO) ≡
FO(Q0, most)

See Peters and Westerståhl (2006), ch. 13.2, for proofs of this and similar facts.

12.
Nor does it have the completeness property or the Tarski property,
though it does have the Löwenheim property.

13.
See Ebbinghaus and Flum (1995) for the mathematics, and
Westerståhl (1989) or Peters and Westerståhl (2006), chs. 13-15, for
surveys focused on linguistic applications.

15.
Similarly, an arbitrary monadic quantifier can be coded as a set of
k-letter words for suitable k.

16.
See, for example, Hopcroft and Ullman (1979) for an introduction to
automata theory.

17.
See van Benthem (1987). ISOM entails that the
automata here are permutation-closed, i.e., that if a binary word is
accepted, so are all permutations of that word; only the number of 1's
and 0's counts.

18.
Here 0 can be defined as the unique y in N such
that y + y = y. For more results in this area, see Mostowski (1998). Clark (2011a) gives an overview of the automata-theoretic perspective of quantifiers, with applications to learning theory.

19.
This is one way of providing for recursive definitions in the
logic. A simpler operator that one can add is the transitive closure
operator. If R is a binary relation, the transitive closure
of R, TC(R), is the smallest transitive
relation containing R. It can also be defined as follows
(cf. the definition of RECIP in the list (15)):

aTC(R)b ⇔ ∃n ≥ 1
∃x0,…,
xn[x0 = a
∧ xn =
b ∧
xiRxi+1
for i < n]

Note the quantification over n: TC(R) is not
in general definable in FO from R. It can also be
defined recursively:

aTC(R)b ⇔ aRb
∨ ∃ x[aRx
∧
xTC(R)b]

To be able to do this inside our logic we can add formulas of the form
TC(x, y,φ)(u, v)
whenever φ is a formula, and the semantic rule that when
a, b ∈ M, ψ(x,
y,[z]) has the free variables shown, and
[c] corresponds to [z],

M ⊨
TC(x, y,ψ(x,
y,[c]))(a, b) ⇔ (a,
b) ∈ TC(ψ(x,
y,[c])M, x,
y)

This gives us the logic FO(TC); the LFP operator
generalizes this to other forms of recursion. See Ebbinghaus and Flum
(1995) for details about the definitions, and for results about these
logics, including those mentioned in this and the next two paragraphs.

20.
Barwise and Cooper (1981), Keenan and Stavi (1986), Higginbotham and
May (1981). But the starting-point was Richard Montague's work in the
late 1960's; notably his paper “The Proper Treatment of
Quantification in Ordinary English” (Montague, 1974), where noun
phrases, including proper names, were treated as quantifiers. The
papers in van Benthem (1986) provided further logical and linguistic
development of these ideas. Surveys of the whole area are
Westerståhl (1989), Keenan and Westerståhl (2011), and
Peters and Westerståhl (2006).

21.
(a) is practically immediate, and for (b) it is easy to verify that
Qrel always satisfies CONSERV and EXT when
Q is of type <1>. In the other direction, any
CONSERV and EXTQ′ has a type <1> ‘counterpart’
Q defined by QM(B) ⇔
Q′M(M, B): then
(Qrel)M(A, B) ⇔
QA(A ∩ B) (by definition of
Qrel) ⇔
Q′A(A, A ∩ B)
(by definition of Q′) ⇔
Q′M(A, A ∩ B)
(by EXT) ⇔
Q′M(A, B) (by CONSERV), so Q′ =
Qrel.

22.
But there again it is often natural to consider the EXT quantifier W rel, where
(W rel)M(A, R)
says that R is a well-ordering of A.

23.
Indeed, the intersective quantifiers mentioned so far have the
stronger property of being cardinal; i.e. only the
cardinality of A ∩ B matters. An example of an
intersective but non-cardinal quantifier
is no _ except Mary, defined in the next
section.

25.
For detailed discussion and further references, see Peters and
Westerståhl (2006): ch. 6.3 for existential there sentences, and
ch. 5 for much more on monotonicity, including the connection with
polarity items.

26.
For much more on this, and on the treatment of possessive and
exceptive determiners in general, see Peters and Westerståhl
(2006), chs. 7 and 8.

27.
We disregard tense here for simplicity, as well as fact that the
plural form indicates that John has more than one book.

28.
See van Benthem (2002) for an informative overview of conditions like
ISOM in the context of logicality. For recent contributions in this area, see Bonnay (2008) and Feferman (2010).

29.
See Keenan and Westerståhl (2011) for more examples and
discussion.

30.
(38) may also have a proportional reading, saying that the
proportion of smokers among women is greater than the proportion of
smokers among men, i.e.,

32.
The last mentioned result is proved by Luosto (2000); the proof is
quite difficult. More general results on the undefinability of
resumption can be found in Hella, Väänänen, and
Westerståhl (1997). Some discussion of the linguistic aspects
appears in Peters and Westerståhl (2002).

33.
This is one of the readings of (41); the other one is
two y(B(y), most x(A(x),
R(x, y))). These quantifiers can be
described as iterations of most and two;
Keenan and Westerståhl (2011) have a detailed account of this
operation as well as a survey of the properties of iterated
quantifiers.

34.
See Dalrymple et al. (1998) for an extended discussion. RECIP is not
definable in FO; indeed it is not definable using monadic
quantifiers at all (Peters and Westerståhl, 2006, ch. 15).

35.Br(Q1, Q2) is EXT if Q1 and
Q2 are, so we assume this and drop the subscript
M.

36.
Branching or partially ordered quantifiers is another way of
generalizing (prefixes of) ∀ and ∃; it appeared in logic
with Henkin (1961), and Hintikka (1973) argued that partially ordered
prefixes with ∀ and ∃ occur essentially in English
too. The debate that followed Hintikka's proposal was re-analyzed by
Barwise (1978), who also suggested that (45) is a clearer example of
branching in English than Hintikka's original examples. Semantically,
branching quantifiers are already subsumed under our notion of a
generalized quantifier, since they can all be seen as polyadic
quantifiers, like Br(Q1,
Q2), although the special syntax is then lost.

But it should also be noted that the construction in (45) only works
in certain cases, namely, when Q1 and
Q2 are right monotone increasing, i.e.,
Qi(A, B) and B
⊆ B′ entails that
Qi(A, B′), i = 1,
2. Note that then

Qi(A, B) ⇔
∃X [Q(A, X) & X
⊆ B]

and one sees that (45) is a generalization of this. There has been
some discussion about if and how (45) can be reformulated for other
quantifiers; apart from Barwise (1978), see Westerståhl (1987)
and Sher (1997).

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